37. Complete the following definitions/statements. a. is an eigenvalue of a squa
ID: 3116753 • Letter: 3
Question
37. Complete the following definitions/statements. a. is an eigenvalue of a square matrix A if b. A linearly independent set of vectors which spans a vector spaceV is calleda of the vector space c. The set vAof vectors is an orthonormal set of vectors if C. The set v,,v d. Any set of k vectors, k 4, in the 3-space R3 is linearly e. The standard basis of Ris f. The set of solutions of the system Ax-0, A is m by n matrix, is a subspace of the -dimensional vector space. The subspace in the above statement is called the g. of Page 4 of 7 h. The row-rank of a matrix is If A is an m by n matrix with linearly independent column vectors then A-QR State clearly what is Q? What is R? i.Explanation / Answer
37.
(a). is an eigenvalue of a square matrix A if det(A- I) = 0.
(b). A linearly independent set of vectors which spans a vector space V is called a basis of the vector space V.
(c ). The set {v1,v2,…,vk} of vectors is an orthonormal set of vectors if vi . vj = 0 for i j, 1 i, j k and if ||vi||=1 for 1 i k.
(d).Any set of k vectors, k 4 in the 3-vector space R3 is linearly dependent.
(e ). The standard basis of R5 is {(1,0,0,0,0)T,(0,1,0,0,0)T,(0,0,1,0,0)T, (0,0,0,1,0)T,(0,0,0,0,1)T}.
(f). The set of solutions to the system Ax = 0, A is a mxn matrix , is a subspace of the n-dimensional vector space.
(g). The subspace in the above statement is called a subspace of Rn.
(h). The row rank of a matrix is equal to the number of non-zero rows in its RREF.
(i). If A is a mxn matrix with linearly independent columns and A = QR, then Q is a mxn matrix with orthonormal columns and R is a n × n, upper triangular matrix with non-zero diagonal elements.
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